Generating function (physics)

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Generating functions which arise in Hamiltonian mechanics are quite different from generating functions in mathematics. In the case of physics, generating functions act as a bridge between two sets of canonical variables.

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[edit] Details

There are four basic generating functions, summarized by the following table.

Generating Function Its Derivatives
F_1 = F_1(q, Q, t) \, p = \frac{\partial F_1}{\partial q} \, and P = - \frac{\partial F_1}{\partial Q} \,
F_2 = F_2(q, P, t) - QP \, p = \frac{\partial F_2}{\partial q} \, and Q = \frac{\partial F_2}{\partial P} \,
F_3 = F_3(p, Q, t) + qp \, q = - \frac{\partial F_3}{\partial p} \, and P = - \frac{\partial F_3}{\partial Q} \,
F_4 = F_4(p, P, t) + qp - QP \, q = - \frac{\partial F_4}{\partial p} \, and Q = \frac{\partial F_4}{\partial P} \,

[edit] Example

Sometimes one can turn a given Hamiltonian into one that looks a bit more like the harmonic oscillator Hamiltonian, which is

H = aP^2 + b Q^2 \,

So, as an example, if one was given the Hamiltonian

H = \frac{1}{2q^2} + \frac{p^2 q^2}{2} \quad \quad \quad \quad (1) \,
(where p is generalized momentum, and q is the generalized coordinate.)

a good canonical transformation to choose would be

P = pq^2 \, and Q = \frac{-1}{q} \quad \quad \quad \quad (2) \,

This turns the Hamiltonian into

H = \frac{Q^2}{2} + \frac{P^2}{2} \,

which is in the form of the harmonic oscillator Hamiltonian.

The generating function, F, for this transformation is of the 3rd kind,

F = F_3(p,Q) .\,

To find F explicitly, use the equation for its derivative (from the table above),

P = - \frac{\partial F_3}{\partial Q} \,

and substitute the expression for P from equation (2), expressed in terms of p and Q:

\frac{p}{Q^2} = - \frac{\partial F_3}{\partial Q} .\,

Integrating this with respect to Q results in an equation for the generating function of the transformation given by equation (2):

F_3(p,Q) = \frac{p}{Q} \,

To confirm that this is the correct generating function, verify that it matches (2):

q = - \frac{\partial F_3}{\partial p} = \frac{-1}{Q} \,

[edit] See also

[edit] References


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